ABSTRACT

Given the inability to foresee all possible scenarios, it is justified to desire an efficient trust-region subproblem solver capable of delivering any desired level of accuracy on demand; that is, the accuracy obtainable for a given trust-region subproblem should not be partially dependent on the problem itself. Current state-of-the-art iterative eigensolvers all fall into the class of restarted Lanczos methods; whereas, current iterative trust-region solvers at best reduce to unrestarted Lanczos methods; which in this context are well known to be numerically unstable with impractical memory requirements. In this paper, we present the first iterative trust region subproblem solver that at its core contains a robust and practical eigensolver. Our solver leverages the recently announced work of Stathopoulos and Orginos which has not been noticed by the optimization community and can be utilized because, unlike other restarted Lanczos methods, its restarts do not necessarily modify the current Lanczos sequence generated by Conjugate Gradient methods (CG). This innovated strategy can be utilized in the context of TR solvers as well. Moreover, our TR subproblem solver adds negligible computational overhead compared to existing iterative TR approaches.

摘要

鉴于无法预见所有可能的情况，有理由期望一个高效的信任域子问题求解器能够按需提供任何所需的准确度水平；也就是说，对于给定的信任域子问题可获得的准确性不应部分取决于问题本身。当前最先进的迭代特征求解器都属于重新启动的 Lanczos 方法类；然而，当前的迭代信任域求解器充其量只能简化为未重新启动的 Lanczos 方法；在这种情况下，众所周知，它在数值上不稳定，具有不切实际的内存要求。在本文中，我们提出了第一个迭代信任域子问题求解器，其核心包含一个强大且实用的特征求解器。我们的求解器利用了最近宣布的 Stathopoulos 和 Orginos 的工作，优化社区尚未注意到这些工作并且可以利用，因为与其他重新启动的 Lanczos 方法不同，它的重新启动不一定会修改由共轭梯度方法 (CG) 生成的当前 Lanczos 序列. 这种创新策略也可以在 TR 求解器的上下文中使用。此外，与现有的迭代 TR 方法相比，我们的 TR 子问题求解器增加的计算开销可以忽略不计。